#382623

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14339
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,10,7,11,6,22)(3,8,13,24,15,25,14,32)(4,16,20,28,19,29,21,9)(12,30,17,27,18,26,31,23);(1,3)(2,8)(4,17,19,31)(5,13)(6,14)(7,15)(9,26,28,30)(10,24)(11,25)(12,20,18,21)(16,23,29,27)(22,32);(1,4)(2,9)(3,12,15,18)(5,21)(6,20)(7,19)(8,23,25,27)(10,29)(11,28)(13,31,14,17)(16,22)(24,26,32,30);(1,3)(2,8)(4,18)(5,13)(6,14)(7,15)(9,27)(10,24)(11,25)(12,19)(16,26)(17,21)(20,31)(22,32)(23,28)(29,30)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289819);